The Gabor system of a function is the set
of all of its integer translations and modulations. The Balian-Low
Theorem states that the Gabor system of a function which is well
localized in both time and frequency cannot form an Riesz basis for
$L^2(\mathbb{R})$.
An important tool in the proof is a characterization of the Riesz basis
property in terms of the boundedness of the Zak transform of the
function. In this talk, we will discuss results showing that weaker
basis-type properties also correspond to boundedness
of the Zak transform, but in the sense of Fourier multipliers. We will
also discuss using these results to prove generalizations of the
Balian-Low theorem for Gabor systems with weaker basis properties, as
well as for shift-invariant spaces with multiple
generators and in higher dimensions.